On Extreme Values in Stationary Weakly Dependent Random Fields

Abstract

The existing literature on extremal types theorems for stationary random processes and fields is, until now, developed under either mixing or “Coordinatewise (Cw)-mixing” conditions. However, these mixing conditions are very restrictives and difficult to verify in general for many models. Due to these limitations, we extend the existing theory, concerning the asymptotic behaviour of the maximum of stationary random fields, to a weaker and simplest to verify dependence condition, called weak dependence, introduced by Doukhan and Louhichi [Stochastic Processes and their Applications 84 (1999): 313–342]. This stationary weakly dependent random fields family includes models such as Bernoulli shifts, chaotic Volterra and associated random fields, under reasonable addition conditions. We mention and check the weak dependence properties of some specific examples from this list, such as: linear, Markovian and LARCH($\infty$) fields. We show that, under suitable weak-dependence conditions, the maximum may be regarded as the maximum of an approximately independent sequence of sub-maxima, although there may be high local dependence leading to clustering of high values. These results on asymptotic max-independence allow us to prove an extremal types theorem and discuss domain of attraction criteria in this framework. Finally, a numerical experiment using a non-mixing weakly dependent random field is performed.